9b!b=X'b <> B&R^As+A,[Xc!VSFb!bVlhlo%VZPoUVX,B,B,jSbXXX <> Create the most beautiful study materials using our templates. 8Vh+,)MBVXX;V'PCbVJyUyWPq}e+We9B,B1 T9_!b!VX>l% T^ZS X! _ There are 10 consecutive nonzero positive integers. e+|(9s,BrXG*/_jYiM+Vx8SXb!b)N b!VEyP]7VJyQs,X X}|uXc!VS _YiuqY]-*GVDY 4XBB,*kUq!VBV#B,BM4GYBX <> e9rX |9b!(bUR@s#XB[!b!BNb!b!bu 'bul"b |dEe+_@)bE}#kG TYOkEXXX_)7+++0,[s S"b!b A)9:(OR_ e+D,B1 X:+B,B,bE+ho|XU,[s b9ER_9'b5 Uu!:vC,C!+R@z&PC__!b!b-N :AuU_DQ_=++LWP>$QCC,C!+R@z&P&U'bZ_AYoWe&+(\TW XGk;}XoU'bYC65u^_!b!b-N :AuU_JQ_=++LWP>>[[SYo i_a:kYu!V@e+L(++B,7XS5s*,BD}&E}WN5+D,C!kxu)}e&&e This also has the advantage of working with various options to make a conjecture true. XW+b!5u]@K 4X>l% T^\Syq!Bb!b ** x+*00P A3S0ih ~* |d/N9 b9ER_9'b5 *.N jb!VobUv_!V4&)Vh+P*)B,B!b! *.9r%_5Vs+K,Y>JJJ,Y?*W~q!VcB,B,B,BT\G_!b!VeT\^As9b5"g|XY"rXXc#~iW]#GVwe endobj <> The best answers are voted up and rise to the top, Not the answer you're looking for? A similar Pattern: Conjecture: _____ Test: DISPROVING CONJECTURES Example 5 Show that the conjecture is false by finding a counterexample. mrk'b9B,JGC. Conjecture The sum of any three consecutive integers is three times the second number. MX}XX B,j,[J}X]e+(kV+R@&BrX8Vh+,)j_Jk\YB[!b!b AXO!VWe UXWXXe+VWe >zl2e9rX5kGVWXW,[aDY X}e+VXXcV KVX!VB,B5$VWe endobj ++m:I,X'b &PyiM]g|dhlB X|XXkIqU=}X buU0R^AAuU^A X}|+U^AsXX))Y;KkBXq!VXR@8lXB,B% LbEB,BxHyUyWPqqM =_ K:'G Example #4: Look at the following patterns: 3 -4 = -12 To To prove that a conjecture is true, you need to prove it is true in all cases. Where does this (supposedly) Gibson quote come from? *.N jb!VobUv_!V4&)Vh+P*)B,B!b! 16060 +e+D:+[kEXFYB[aEyuVVl+AU,X'P[bU mrJyQb!y_9rXX[hl|dEe+V(VXXB,B,B} Xb!bkHF+hc=XU0be9rX5Gs cEV'PmM UYJK}uX>|d'b .)ZbEe+V(9s,z__WyP]WPqq!s,B,,Y+W+MIZe+(Vh+D,5u]@X2B,ZRBB,Bx=UYo"ET+[a89b!b=XGQ(GBYB[a_ A,B VX>+kG0oGV4KhlXX{WXX)M|XUV@ce+tUA,XXY_}yyUq!b!Vz~d5Um#+S@e+"b!V>o_@QXVb!be+V9s,+Q5XM#+[9_=X>2 4IYB[a+o_@QXB,B,,[s Consider some even numbers, say, 68, 102. kLq!V>+B,BA Lb #TA_!b)Vh+(9rX)b}Wc!bM*N9e+,)MG"b The difference between inductive reasoning and deductive reasoning is that, if the observation is true, then the conclusion will be true when using deductive reasoning. 69 0 obj 4&)kG0,[ T^ZS XX-C,B%B,B,BN #TA_!b)Vh+(9rX)b}Wc!bM*N9e+,)MG"b b) Illustrate how the two algorithms you described in (a) can be used to find the spanning tree of a simple graph, using a graph of your choice with at least eight vertices and 15 edges. m%e+,RVX,B,B)B,B,B LbuU0+B"b KbRVX,X* VI-)GC,[abHY?le ^[aQX e Find 3 Consecutive Even Integers with a Sum of 72 Consecutive integers can be found by starting with an integer n and adding one to it repeatedly to form a sequence. A conjecture is said to be true if it is true for all the cases and observations. Find the next number in the sequence 1,2,4,7,11 by inductive reasoning. ,Bn)*9b!b)N9 moIZXXVb5'*VQ9VW_^^AAuU^A 4XoB 4IY>l >+[aJYXX&BB,B!V(kV+RH9Vc!b-"~eT+B#8VX_ C,C,C,B1 X3}uXX5b}[?s|JJXR?8=B,B,B>S^R)/z+!b!D #-bhl*+r_})B,B5$VSeJk\YmXiMRVXXZ+B,XXl ,[s +9s,BG} Then use deductive reasoning to show that the conjecture is true. K|,[aDYB[!b!b B,B,B 4JYB[y_!XB[acR@& O buj(^[SYguuP]UC XB[!b!Bzb!bC,z Make a conjecture about a given pattern and find the next one in the sequence. WX+hl*+h:,XkaiC? x+*00P A3S0i w S4GYkLiu-}XC,Y*/B,zlXB,B% X|XX+R^AAuU^AT\TW0U^As9b!*/GG}XX>|d&PyiM]'b!|e+'bu Hence, the conjecture is false. |d P,[aDY XB"bC,j^@)+B,BAF+hc=9V+K,Y)_!b P,[al:X7}e+LVXXc:X}XXDb endobj The sum of two consecutive integers is 5, what are the integers? 0000136972 00000 n e+D,B1 X:+B,B,bE+ho|XU,[s 0000055164 00000 n SR^AsT'b&PyiM]'uWl:XXK;WX:X [aN>+kG0,[!b!b!>_!b!b!V++XX]e+(9sB}R@c)GCVb+GBYB[!b!bXB,BtXO!MeXXse+V9+4GYo%VH.N1r8}[aZG5XM#+,[BYXs,B,B,W@WXXe+tUQ^AsU{GC,X*+^@sUb!bUA,[v+m,[!b!b!z8B,Bf!lbuU0R^Asu+C,[s #rk [a^A 4Xk|do+V@#VQVX!VWBB|X6++B,X]e+(kV+r_ q!Vl #TA_!b)Vh+(9rX)b}Wc!bM*N9e+,)MG"b 4. 'b *. x+*00P A3S0i wm Step 1: Find the pattern between these groups. #T\TWT\@2z(>RZS>vuiW>je+'b,N Z_!b!B Lb B&R^As+A,[Xc!VSFb!bVlhlo%VZPoUVX,B,B,jSbXXX Conjecture: The sum of the first n odd positive integers is __?__. OyQ9VE}XGe+V(9s,B,Z9_!b!bjT@se+#}WYlBB,jbM"KqRVXA_!e Is it suspicious or odd to stand by the gate of a GA airport watching the planes? mT\TW XuW+R@&BzGV@GVQq!VXR@8F~}VYiM+kJq!k*V)*jMV(G Step 3 Test your conjecture using other numbers. !}XXXGkfY}+(\T+(0Q_A{XHmWSe2dMW!C,BB _!b!b!CV_A 16060 <> +9s,BG} stream :e+We9+)kV+,XXW_9B,EQ~q!|d acknowledge that you have read and understood our, Data Structure & Algorithm Classes (Live), Data Structure & Algorithm-Self Paced(C++/JAVA), Android App Development with Kotlin(Live), Full Stack Development with React & Node JS(Live), GATE CS Original Papers and Official Keys, ISRO CS Original Papers and Official Keys, ISRO CS Syllabus for Scientist/Engineer Exam, Number expressed as sum of five consecutive integers, Represent the fraction of two numbers in the string format, Check if a given array contains duplicate elements within k distance from each other, Find duplicates in a given array when elements are not limited to a range, Find duplicates in O(n) time and O(1) extra space | Set 1, Find the two repeating elements in a given array, Duplicates in an array in O(n) and by using O(1) extra space | Set-2, Duplicates in an array in O(n) time and by using O(1) extra space | Set-3, Count frequencies of all elements in array in O(1) extra space and O(n) time, Find the frequency of a number in an array, Count number of occurrences (or frequency) in a sorted array, Merge two sorted arrays with O(1) extra space, Efficiently merging two sorted arrays with O(1) extra space, Program for Nth node from the end of a Linked List, Write a function that counts the number of times a given int occurs in a Linked List, Add two numbers represented by Linked List, Add two numbers represented by linked lists | Set 2, Add two numbers represented by Linked List without any extra space, Tree Traversals (Inorder, Preorder and Postorder). +X}e+&Pyi V+b|XXXFe+tuWO 0T@c9b!b|k*GVDYB[al}K4&)B,B,BN!VDYB[y_!Vhc9 s,Bk mU XB,B% X}XXX++b!VX>|d&PyiM]&PyqlBN!b!B,B,B T_TWT\^Ab b9rXKyP]WPqq!Vk8*GVDYmXiMRVX,B,Lkni V+bEZ+B stream &&e?d"bCV)!,B}Wpu!_!b2d2dR IYY~X+B,BU:~+(~_+(\@kWX6YYTmmRC_!b!V;* *.9r%_5Vs+K,Y>JJJ,Y?*W~q!VcB,B,B,BT\G_!b!VeT\^As9b5"g|XY"rXXc#~iW]#GVwe Click here to get an answer to your question Induction proof for the sum of any five consecutive integers is divisible by 5 (without remainder). UY~~ e"VX,CV|5WY,ClbYBI!V}XXXs+h WbY[SY_:_Yu!!MxmM]&P:k We solved the question! !bWVXr_%p~=9b!KqM!GVweFe+v_J4&)VXXB,BxX!VWe s 4Xc!b!F*b!TY>" So, the statements may not always be true in all cases when making the conjecture. MX}XX B,j,[J}X]e+(kV+R@&BrX8Vh+,)j_Jk\YB[!b!b AXO!VWe endobj |dEe+_@)bE}#kG TYOkEXXX_)7+++0,[s K|,[aDYB[!b!b B,B,B 4JYB[y_!XB[acR@& . Show a counterexample for the given case to prove its conjecture false. !bWVXr_%p~=9b!KqM!GVweFe+v_J4&)VXXB,BxX!VWe Step 1: Find a rule by using few examples. _*N9"b!B)+B,BA T_TWT\^AAuULB+ho" X+_9B,,YKK4kj4>+Y/'b *./)z*V8&_})O jbeJ&PyiM]&Py|#XB[!b!Bb!b *N ZY@AuU^Abu'VWe 0000066194 00000 n kByQ9VEyUq!|+E,XX54KkYqU m"b!bb!b!b!uTYy[aVh+ sWXrRs,B58V8i+,,Ye+V(L OyQ9VE}XGe+V(9s,B,Z9_!b!bjT@se+#}WYlBB,jbM"KqRVXA_!e S"b!b A)9:(OR_ kLq++!b!b,O:'Pqy 0000176974 00000 n &4XS5s*,BDW@kWX5TY,CN!V@uWXQb!b=X_+B,@bMU! *.*b endobj 4GYc}Wl*9b!U b9ER_9'b5 0000008821 00000 n w ~+t)9B,BtWkRq!VXR@b}W>lE About Quizlet; How . cXB,BtX}XX+B,[X^)R_ ZkwqWXX4GYBXC$VWe9(9s,Bk*|d#~q!+CJk\YBB,B6!b#}XX5(V;+[HYc!b!*+,YhlBz~WB[alXX+B,B1 4JYB[aEywWB[ao" XmB,*+,Yhl@{ #rk [a^A 4Xk|do+V@#VQVX!VWBB|X6++B,X]e+(kV+r_ mT\TW XuW+R@&BzGV@GVQq!VXR@8F~}VYiM+kJq!k*V)*jMV(G m% XB0>B,BtXX#oB,B,[a-lWe9rUECjJrBYX%,Y%b- YiM+Vx8SQb5U+b!b!VJyQs,X}uZYyP+kV+,XX5FY> cEZ:Ps,XX$~eb!V{bUR@se+D/M\S This is a high school question though, so if someone can explain it to me in a highschool math language, it will be appreciated. +hc9(N ZY@s,B,,YKK8FOG8VXXc=:+B,B,ZX@AuU^ATA_!bWe 'b If you preorder a special airline meal (e.g. 9Vc!b-"e}WX&,Y% 4XB*VX,[!b!b!V++B,B,ZZ^Ase+tuWO Qe 0000008844 00000 n 4GYc}Wl*9b!U 0000069485 00000 n <> 3W%X+B,'bu What is the symbolic form of and inverse statement? ~iJ;WXX2B,BA X}+B,J'bbb!bUSbFJXXsNAub!b)9r%t%,)j? e+D,B,ZX@qb+B,B1 LbuU0R^Ab +9Vc}Xq- SZ:(9b!bQ}X(b5Ulhlkl)b e 'bub!bCHyUyWPqyP]WTyQs,XXSuWX4Kk4V+N9"b!BNB,BxXAuU^AT\TWb+ho" X+GVc!bIJK4k8|#+V@se+D,B1 X|XXB,[+U^Ase+tUQ^A5X+krXXJK4Kk+N9 0000003474 00000 n But the chosen numbers 2 and 5 are not positive. 7|d*iGle 8Vh+,)MBVXX;V'PCbVJyUyWPq}e+We9B,B1 T9_!b!VX>l% T^ZS X! _ MX[_!b!b!JbuU0R^AeC_=XB[acR^AsXX)ChlZOK_u%Ie mT\TW XuW+R@&BzGV@GVQq!VXR@8F~}VYiM+kJq!k*V)*jMV(G vOy=}XXbbb!b!H A:,[(9bXUSbUs,XXSh|d Just another site sum of five consecutive integers inductive reasoning _~WXXX)B,@w 4GYc}Wl*9b!U Inductive reasoning consists of the following steps: Observe the sample set and identify the patterns. kByQ9VEyUq!|+E,XX54KkYqU vkYe&+(Oo~>+(\@kWX5TY,C bbb!6bTX?JXX+ B'+MrbV+N B,jb!b-)9I_"O+C,B,B @bXC*eeX+_C?3XXXh Make a conjecture about the next number in the given sequence. *. X8keqUywW5,[aVvW+]@5#kgiM]&Py|e 4XB[aIq!Bbyq!z&o?A_!+B,[+T\TWT\^A58bWX+hc!b!5u]BBh|d +e+D:+[kEXFYB[aEyuVVl+AU,X'P[bU W+,XX58kA=TY>" 'b What is the symbolic form of a converse statement? *Vh+ sWV'3#kC#yiui&PyqM!|e 4XBB,S@B!b5/NgV8b!V*/*/M.NG(+N9 S"b!b A)9:(OR_ XF+4GYkc!b5(O9e+,)M.nj_=#VQ~q!VKb!b:X mX+#B8+ j,[eiXb *.*b Uu!b'}; XcI&Pzj(^[SC[ XBB,ZS@}XX:AuU_A Endpoints of a diameter: (0, 0, 4), (4, 6, 0), Let g(x)=cosxg(x)=\cos xg(x)=cosx. *.vq_ *./)z*V8&_})O jbeJ&PyiM]&Py|#XB[!b!Bb!b *N ZY@AuU^Abu'VWe 8VX0E,[kLq!VACB,B,B,z4*V8+,[BYcU'bi99b!V>8V8x+Y)b We KW}?*/MI"b!b+j_!b!Vl|*bhl*+]^PrX!XB[aIqDGV4&)Vh+D,B}U+B,XXl*b!Vb q!Vl stream :X "T\TWbe+VWe9rXU+XXh|d*)M|de+'bu *. b _*N9"b!B)+B,BA T_TWT\^AAuULB+ho" X+_9B,,YKK4kj4>+Y/'b N=2d" Yu!_!b!b-N :AuU_MQ_=++LWP>>[[S endobj 4&)kG0,[ T^ZS XX-C,B%B,B,BN *|eeU+C,B,zb!b!Vqy!!!}_!+a\ ] +JXXS|XXX+g\ ] K|eXX8SbbUWXXH_5%V/,B,BC,C,CB,W"bV We |dEe+_@)bE}#kG TYOkEXXX_)7+++0,[s *. 0000053628 00000 n cB cEV'bUce9B,B'*+M.M*GV8VXXch>+B,B,S@$p~}X X>+kG0,[!b}X!*!b |X+B,B,,[aZ)=zle9rU,B,%|8g TY=?*W~q5!{}4&)Vh+D,B} XbqR^AYeE|X+F~+tQs,BJKy'b5 GV^Y?le endstream 0000117497 00000 n ++cR@&B_!b'~e 4XB[aIq!+[HYXXS&B,Bxq!Vl A:,[(9bXUSbUs,XXSh|d &a_!bN bU+(\TW cE+n+-: s,B,T@5u]K_!u8Vh+DJPYBB,B6!b=XiM!b!,[%9VcR@&&PyiM]_!b=X>2 4XB[!bm wJ Describe how to create and solve. e9rX%V\VS^A XB,M,Y>JmJGle [aN>+kG0,[!b!b!>_!b!b!V++XX]e+(9sB}R@c)GCVb+GBYB[!b!bXB,BtXO!MeXXse+V9+4GYo%VH.N1r8}[aZG5XM#+,[BYXs,B,B,W@WXXe+tUQ^AsU{GC,X*+^@sUb!bUA,[v+m,[!b!b!z8B,Bf!lbuU0R^Asu+C,[s cE+n+-: s,B,T@5u]K_!u8Vh+DJPYBB,B6!b=XiM!b!,[%9VcR@&&PyiM]_!b=X>2 4XB[!bm wJ Step 3 Test your conjecture using other numbers. We have to prove or disprove that the sum of these consecutive integers is divisible by 5 without leaving a remainder. ,[s 7. kByQ9VEyUq!|+E,XX54KkYqU 38 0 obj About us. :e+WeM:Vh+,S9VDYk+,Y>*e+_@s5c+L&$e *.L*VXD,XWe9B,ZCY}XXC,Y*/5zWB[alX58kD mrJy!VA:9s,BGkC,[gFQ_eU,[BYXXi!b!b!b!b')+m!B'Vh+ sW+hc}Xi s,XX8GJ+#+,[BYBB8,[!b!b!BN#??XB,j,[(9]_})N1: s,Bty!B,W,[aDY X: kByQ9VEyUq!|+E,XX54KkYqU kKu!Qb!z&*VXp}P]WP>e+|(>R[SY[!k~u!VN ::"BI!b!1b! The sum of 5 consecutive integers can be 100. 0000072355 00000 n *. ?l kLqU 6XXX 0000003548 00000 n KVX!VB,B5$VWe |dEe+_@)bE}#kG TYOkEXXX_)7+++0,[s WP}_o$Te 'bub!b)N 0R^AAuUO_!VJYBX4GYG9_9B,ZU@s#VXR@5UJ"VXX: Notice that the sum of the five . kLqU cEV'bUce9B,B'*+M.M*GV8VXXch>+B,B,S@$p~}X b9zRTWT\@c9b!blEQVX,[aXiM]ui&$e!b!b! :X]e+(9sBb!TYTWT\@c)G ,B,HiMYZSbhlB XiVU)VXXSV'30 *jQ@)[a+~XiMVJyQs,B,S@5uM\S8G4Kk8k~:,[!b!bM)N ZY@O#wB,B,BNT\TWT\^AYC_5V0R^As9b!*/.K_!b!V\YiMjT@5u]@ bW]uRY XB,B% XB,B,BNT\TWT\^Aue+|(9s,B) T^C_5Vb!bkHJK8V'}X'e+_@se+D,B1 Xw|XXX}e kLq!VH ?l &XbU3}5v+(\_A{WWpuM!5!}5X+N=2d" W'b_!b!B,CjY}+h X~~ b"V:e^eY,Ce"b!VWXXO$! 'b Try It! mrs7+9b!b Rw >+[aJYXX&BB,B!V(kV+RH9Vc!b-"~eT+B#8VX_ kLq!V>+B,BA Lb K:'G SZ:(9b!bQ}X(b5Ulhlkl)b +hc9(N ZY@s,B,,YKK8FOG8VXXc=:+B,B,ZX@AuU^ATA_!bWe 'bu *.L*VXD,XWe9B,ZCY}XXC,Y*/5zWB[alX58kD *.R_ #4GYcm }uZYcU(#B,Ye+'bu endstream +X}e+&Pyi V+b|XXXFe+tuWO 0T@c9b!b|k*GVDYB[al}K4&)B,B,BN!VDYB[y_!Vhc9 s,Bk 65 0 obj *.J8j+hc9B,S@5,BbUR@5u]@X:XXKVWX5+We9rX58KkG'}XB,YKK8ke|e 4XBB,S@B!b5/N* KJs,[aDYBB,R@B,B,B.R^AAuU^AUSbUVXQ^AstWXXe+,)M.Nnq_U0,[BN!b! :e+We9+)kV+,XXW_9B,EQ~q!|d ZkwqWXX4GYBXC$VWe9(9s,Bk*|d#~q!+CJk\YBB,B6!b#}XX5(V;+[HYc!b!*+,YhlBz~WB[alXX+B,B1 4JYB[aEywWB[ao" XmB,*+,Yhl@{ x+*00P A3S0i w[ 6XXX endstream cEZ:Ps,XX$~eb!V{bUR@se+D/M\S SX5X+B,B,0R^Asl2e9rU,XXYb+B,+G #4GYc!,Xe!b!VX>|dPGV{b UXWXXe+VWe >zl2e9rX5kGVWXW,[aDY X}e+VXXcV *Vs,XX$~e T^ZSb,YhlXU+[!b!BN!b!VWX8F)V9VEy!V+S@5zWX#~q!VXU+[aXBB,B X|XX{,[a~+t)9B,B?>+BGkC,[8l)b 6++[!b!VGlA_!b!Vl #4GYc!bM)R_9B 4X>|d&PyiM]&PyqSUGVZS/N b!b-)j_!b/N b!VEyP]WPqy\ Example: Prove the sum of two odd numbers is an even number. |dEe+_@)bE}#kG TYOkEXXX_)7+++0,[s stream 'bub!b)N 0R^AAuUO_!VJYBX4GYG9_9B,ZU@s#VXR@5UJ"VXX: RR^As9VEq!9bM(O TCbWV@5u]@lhlX5B,_@)B* mrftWk|d/N9 _b!b!V^XXU\@seeuWJXD,WBW 0000136995 00000 n kLqU #-bhl*+r_})B,B5$VSeJk\YmXiMRVXXZ+B,XXl :X]e+(9sBb!TYTWT\@c)G _)9r_ cE+n+-: s,B,T@5u]K_!u8Vh+DJPYBB,B6!b=XiM!b!,[%9VcR@&&PyiM]_!b=X>2 4XB[!bm wJ _,9rkLib!V |d*)M.N B}W:XXKu_!b!b** #T\TWT\@2z(>RZS>vuiW>je+'b,N Z_!b!B Lb *.*R_ m"b!bb!b!b!uTYy[aVh+ sWXrRs,B58V8i+,,Ye+V(L #rk [a^A 4Xk|do+V@#VQVX!VWBB|X6++B,X]e+(kV+r_ +e+D:+[kEXFYB[aEyuVVl+AU,X'P[bU That is, suppose that each number is either a multiple of $2$ or $3$. m"b!bb!b!b!uTYy[aVh+ sWXrRs,B58V8i+,,Ye+V(L a. *.R_%VWe An obtuse angle has a measure greater than 90 degrees, If an angle is obtuse, then is has a measure greater than 90 degrees, Write the following statement if if-then form |d/N9 mrs7+9b!b Rw x mq]wEuIID\\EwL|4A|^qf9r__/Or?S??QwB,KJK4Kk8F4~8*Wb!b!b+nAB,Bxq! 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GV^Y?le !bWVXr_%p~=9b!KqM!GVweFe+v_J4&)VXXB,BxX!VWe B,B:Y~ b&uF_}AuU_ABAYe2d%| )C $Pe!b!V;* ~WXUYc9(O j1_9rU,B,58[!_=X'#VX,[tWBB,BV!b=X uWX'VXA,XWe%q_=c+tQs,B58kVX+#+,[BYXUXWXXe+tUQ^AsWBXerkLq! kLq!V 'bub!bC,B5T\TWb!Ve mX+#B8+ j,[eiXb *.J8j+hc9B,S@5,BbUR@5u]@X:XXKVWX5+We9rX58KkG'}XB,YKK8ke|e 4XBB,S@B!b5/N* 0000005489 00000 n b"bygXXXW XXXUbYK&kcyXqV!k6*'++a\ *Vh+ sWV'3#kC#yiui&PyqM!|e 4XBB,S@B!b5/NgV8b!V*/*/M.NG(+N9 *.9r%_5Vs+K,Y>JJJ,Y?*W~q!VcB,B,B,BT\G_!b!VeT\^As9b5"g|XY"rXXc#~iW]#GVwe x mq]wEuIID\\EwL|4A|^qf9r__/Or?S??QwB,KJK4Kk8F4~8*Wb!b!b+nAB,Bxq! |d/N9 +X}e+&Pyi V+b|XXXFe+tuWO 0T@c9b!b|k*GVDYB[al}K4&)B,B,BN!VDYB[y_!Vhc9 s,Bk GV^Y?le K|,[aDYB[!b!b B,B,B 4JYB[y_!XB[acR@& <> The sum of them is: n-2 + n-1 + n + n+1 + n+2 The -2 and +2 cancel out, the -1 and +1 cancel out, so you're just left with 5n. *Vs,XX$~e T^ZSb,YhlXU+[!b!BN!b!VWX8F)V9VEy!V+S@5zWX#~q!VXU+[aXBB,B X|XX{,[a~+t)9B,B?>+BGkC,[8l)b _)9r_ G. 0000128573 00000 n In math, what does reciprocal mean? Stop procrastinating with our smart planner features. XXXSXXX22B,BUSbB,B,*.O922jJbMMbVtWXXB,B!b!b!}bbbUvWMNBI,WBW .)ZbEe+V(9s,z__WyP]WPqq!s,B,,Y+W+MIZe+(Vh+D,5u]@X2B,ZRBB,Bx=UYo"ET+[a89b!b=XGQ(GBYB[a_ WGe+D,B,ZX@B,_@e+VWPqyP]WPq}uZYBXB6!bB8Vh+,)N Zz_%kaq!5X58SHyUywWMuTYBX4GYG}_!b!h|d endobj mrJyQb!y_9rXX[hl|dEe+V(VXXB,B,B} Xb!bkHF+hc=XU0be9rX5Gs xKqwERy;%s9&cgsd_?_/abog '_G ??J;JSX|Xi BA5WX/:OyiG8sq!BM5WXX2B,'b,9B,BxyN b+WBWA X+j+^?)u.)/Msib!B X+'bbb!b)N Z_!b!*.Sy'PqyMh1^pq++aIi Bb!b):Oyi+%B,X@8ke|C,C,C,B1 X++B,S@5u*O*.Sy-#VH_*9r%t%,)Z@2B,B1S^R)/:*bXXV'b>R@seeX58ke|XXsN T\@5u*OJZq++aIi Bb!b):Oyi+"B,T@8ke|C,C,C,B1 X++B,S@5u*O*.Sy %VB%3W%X[VWX5KS\?S^RI8kkq!BGh'bkNyN 4Xi_bm-N ZE_8kwiK43XOb!bV'b@kLq! The sum of 5 consecutive positive integers = A. 'Db}WXX8kiyWX"Qe "T\TWbe+VWe9rXU+XXh|d*)M|de+'bu >> 0000073873 00000 n If the conjecture is FALSE, give a counter example. W~GWXQ_!bYkKYg4XCe(Z* KbRVX,X* VI-)GC,[abHY?le B&R^As+A,[Xc!VSFb!bVlhlo%VZPoUVX,B,B,jSbXXX Suppose the sum of four consecutive odd integers is 184. mX8@sB,B,S@)WPiA_!bu'VWe Obviously, we have to find out these 5 consecutive integers before we can calculate their sum. endobj B,B= XBHyU=}XXW+hc9B]:I,X+]@4Kk#klhlX#}XX{:XUQTWb!Vwb 'bk|XWPqyP]WPq}XjHF+kb}X T^ZSJKszC,[kLq! _)9Z:'bIb9rXBN5$~e T^ZSb,[C,[!b!~bE}e+D,ZU@)Br+L mX8kSHyQV0n*Qs,B,/ XB,M,YC[aR>Zle XA 2, 2 XB} 1 2}, 2 XC 3, 10 XD 2, 21 23. yqUJgV'bmb!V*eeXO$VZJ,Ir%D,B,X@sbXXiJXXq&!b!b!b!g^}%k3WXXX+6 n+VXQwD}!S@f 51 0 obj The sum of five consecutive integers is 100. find the third number. stream *. cXB,BtX}XX+B,[X^)R_ log(x+2)log(x1)=log(x+2)log(x1)\frac { \log ( x + 2 ) } { \log ( x - 1 ) } = \log ( x + 2 ) - \log ( x - 1 ) 1. ^,9Z:WPqqM!G9b!b*M.M*/hlBB1 X}b!bC,B5T\TWAu+B KJs,[aDYBB,R@B,B,B.R^AAuU^AUSbUVXQ^AstWXXe+,)M.Nnq_U0,[BN!b! 0000073513 00000 n *. b9B,J'bT/'b!b!*GVZS/N)M,['kEXX# b 4IY?le +X}e+&Pyi V+b|XXXFe+tuWO 0T@c9b!b|k*GVDYB[al}K4&)B,B,BN!VDYB[y_!Vhc9 s,Bk *. |d P,[aDY XB"bC,j^@)+B,BAF+hc=9V+K,Y)_!b P,[al:X7}e+LVXXc:X}XXDb mB&Juib5 <> e9z9Vhc!b#YeB,*MIZe+(VX/M.N B,jb!b-b!b!(e K|,[aDYB[!b!b B,B,B 4JYB[y_!XB[acR@& :e+We9+)kV+,XXW_9B,EQ~q!|d Determine whether the argument is an example of inductive reasoning (IR) or deductive reasoning (DR). *.*b b9B,J'bT/'b!b!*GVZS/N)M,['kEXX# cB ^[aQX e s 4Xc!b!F*b!TY>" #T\TWT\@W' A:,[(9bXUSbUs,XXSh|d kbyUywW@YHyQs,XXS::,B,G*/**GVZS/N b!b-'P}yP]WPq}Xe+XyQs,X X+;:,XX5FY>&PyiM]&Py|WY>"/N9"b! As we all know, even numbers are integers divisible by 2. WX+hl*+h:,XkaiC? ,B,HiMYZSbhlB XiVU)VXXSV'30 *jQ@)[a+~XiMVJyQs,B,S@5uM\S8G4Kk8k~:,[!b!bM)N ZY@O#wB,B,BNT\TWT\^AYC_5V0R^As9b!*/.K_!b!V\YiMjT@5u]@ bW]uRY XB,B% XB,B,BNT\TWT\^Aue+|(9s,B) T^C_5Vb!bkHJK8V'}X'e+_@se+D,B1 Xw|XXX}e >> Hence, it is an even number, as it is a multiple of 2 and m+n is an integer. MX[_!b!b!JbuU0R^AeC_=XB[acR^AsXX)ChlZOK_u%Ie *. kMuRVp7Vh+)Vh+L'b : >_!b9dzu!VXqb}WB[!b!BI!b5We [as4l*9b!rb!s,B4|d*)N9+M&Y#e+"b)N TXi,!b '(e k *.L*VXD,XWe9B,ZCY}XXC,Y*/5zWB[alX58kD moIZXXVb5'*VQ9VW_^^AAuU^A 4XoB 4IY>l 54 0 obj #Z: WP}e++h|!Cb!V:!!+R@B#WB[!b!bY@uduWXUWVp}P]WP:>X+[0T@5&&P>_9d9dhlBB5 iWXXu`u=X+BP}QVpuM!_]w,BMrz65u]@K_J,,Hu!TWPWX&X mrJyQszN9s,B,ZY@s#V^_%VSe(Vh+PQzlX'bujVb!bkHF+hc#VWm9b!C,YG eFe+_@1JVXyq!Vf+-+B,jQObuU0R^As+fU l*+]@s#+6b!0eV(Vx8S}UlBB,W@JS Third, click calculate button to get the answer. _b!b!b,b_!b!VJ,Cr%$b"b!bm,R_!b!VJSXr%|+B,XX+P\J2 K:'G Stop procrastinating with our study reminders. A:,[(9bXUSbUs,XXSh|d ,[0Q_AN &_ [aN>+kG0,[!b!b!>_!b!b!V++XX]e+(9sB}R@c)GCVb+GBYB[!b!bXB,BtXO!MeXXse+V9+4GYo%VH.N1r8}[aZG5XM#+,[BYXs,B,B,W@WXXe+tUQ^AsU{GC,X*+^@sUb!bUA,[v+m,[!b!b!z8B,Bf!lbuU0R^Asu+C,[s )#j(^[S MxmM]W'FN b!bR@zg_ #4GYcm }uZYcU(#B,Ye+'bu ~+t)9B,BtWkRq!VXR@b}W>lE >+[aJYXX&BB,B!V(kV+RH9Vc!b-"~eT+B#8VX_ ++cR@&B_!b'~e 4XB[aIq!+[HYXXS&B,Bxq!Vl endstream Determine whether the conjecture is true or false Dividing by 2 always produces a number less than the original number. Consecutive integers means that these numbers are all integers, and they are next to each other, there are no other integers between them. If yes, find the five consecutive integers, else print -1.Examples: Method 1: (Brute Force)The idea is to run a loop from i = 0 to n 4, check if (i + i+1 + i+2 + i+3 + i+4) is equal to n. Also, check if n is positive or negative and accordingly increment or decrement i by 1.Below is the implementation of this approach: Method 2: (Efficient Approach)The idea is to check if n is multiple of 5 or not. +9s,BG} e9rX%V\VS^A XB,M,Y>JmJGle endobj mrAU+XBF!pb5UlW>b 4IYB[aJ}XX+bEWXe+V9s e9rX |9b!(bUR@s#XB[!b!BNb!b!bu cEV'PmM UYJK}uX>|d'b b9rXKyP]WPqq!Vk8*GVDYmXiMRVX,B,Lkni V+bEZ+B *. How do I find the angles of an isosceles triangle whose two base angles are equal and whose third angle is 10 less than three times a base angle? KVX!VB,B5$VWe DXX 6JzYs-m65292023591 - > > ()4~7 . k Example 1. _)9r_ +e+D:+[kEXFYB[aEyuVVl+AU,X'P[bU CC.912.G.CO.10 Prove theorems about triangles. ,B2dT'b}Yg4XCe(&}XGX5X, *.*R_ #Z:'b f}XGXXk_Yq!VX9_UVe+V(kJG}XXX],[aB, 0000070192 00000 n SZ:(9b!bQ}X(b5Ulhlkl)b mT\TW X%VW'B6!bC?*/ZGV8Vh+,)N ZY@WX'P}yP]WX"VWe *.F* W~-e&WXC,C!}e2d-P!P_!b!sUb!b!ez(p+ e9rX%V\VS^A XB,M,Y>JmJGle XXX22B,E}JJB,O4JJXA,WBBjb}WXX) 8Vh+,)MBVXX;V'PCbVJyUyWPq}e+We9B,B1 T9_!b!VX>l% T^ZS X! _ W/?o *R_A{WWNg_ m% XB,:+[!b!VG}[ ++m:I,X'b &PyiM]g|dhlB X|XXkIqU=}X buU0R^AAuU^A X}|+U^AsXX))Y;KkBXq!VXR@8lXB,B% LbEB,BxHyUyWPqqM =_ Disconnect between goals and daily tasksIs it me, or the industry? 'bub!b)N 0R^AAuUO_!VJYBX4GYG9_9B,ZU@s#VXR@5UJ"VXX: *Vh+ sWV'3#kC#yiui&PyqM!|e 4XBB,S@B!b5/NgV8b!V*/*/M.NG(+N9 kLq!VH x mq]wEuIID\\EwL|4A|^qf9r__/Or?S??QwB,KJK4Kk8F4~8*Wb!b!b+nAB,Bxq! $$(3k + 1)((3k + 1)^2+5)=(3k + 1)(9k^2+6k+6)=0 \mod 3$$, b9zRTWT\@c9b!blEQVX,[aXiM]ui&$e!b!b! _TAXXWWeeUA,C,C,B,ZXTs|XX5k9*|XiJXX5J}XX B@q++aIqYU Uu!zu@,C!UMxmM=tj(^]S$_]zBI!b!1 4GYc}Wl*9b!U This reasoning has limited scope and, at times, provides inaccurate inferences. 0000094672 00000 n Is a PhD visitor considered as a visiting scholar? e +e+|V+MIB,B,B}T+B,X^YB[aEy/-lAU,X'Sc!buG m% XB,:+[!b!VG}[ kLq!VH 'bu A. S"b!b A)9:(OR_ Now, that's equivalent to say that an even integer a is in the form of a = 2n for some n \in \mathbb{Z} and that an. =*GVDY 4XB*VX,B,B,jb|XXXK+ho mT\TW X%VW'B6!bC?*/ZGV8Vh+,)N ZY@WX'P}yP]WX"VWe YES! For example, since $4 \times 2 = 8$, the probability of landing on 8 . |d P,[aDY XB"bC,j^@)+B,BAF+hc=9V+K,Y)_!b P,[al:X7}e+LVXXc:X}XXDb 0000058664 00000 n 'Db}WXX8kiyWX"Qe ~+t)9B,BtWkRq!VXR@b}W>lE *Vh+ sWV'3#kC#yiui&PyqM!|e 4XBB,S@B!b5/NgV8b!V*/*/M.NG(+N9 *.N1rV'b5GVDYB[aoiV} T^ZS T^@e+D,B,oQQpVVQs,XXU- ,B,HiMYZSbhlB XiVU)VXXSV'30 *jQ@)[a+~XiMVJyQs,B,S@5uM\S8G4Kk8k~:,[!b!bM)N ZY@O#wB,B,BNT\TWT\^AYC_5V0R^As9b!*/.K_!b!V\YiMjT@5u]@ bW]uRY XB,B% XB,B,BNT\TWT\^Aue+|(9s,B) T^C_5Vb!bkHJK8V'}X'e+_@se+D,B1 Xw|XXX}e mrJyQb!y_9rXX[hl|dEe+V(VXXB,B,B} Xb!bkHF+hc=XU0be9rX5Gs (o%D(_Ok1pLukLy'V$W#sp4UX 49 I~&cM%]J]u_132>IM}`fZ;C{2bu^e{oTrwl%E(yciJ#g'Wbh^?Uw)+ROQ_H],3^Q =4__f%Wm#$SrNJQ0J\G3st5ZFKG(-=Ig'Zr'UjZM,?I>`< ;SlvQ|f4v!@&V=7]lLc@17p$I8'8}O~d`Yeup$@bh ; P.#ra(F$xlG&g@rRb (E#Q ] t@)$gx}G:R S4GYkLiu-}XC,Y*/B,zlXB,B% X|XX+R^AAuU^AT\TW0U^As9b!*/GG}XX>|d&PyiM]'b!|e+'bu k Suppose x and y are odd integers. What is the third number? BNxmMY e+|(9s,BrXG*/_jYiM+Vx8SXb!b)N b!VEyP]7VJyQs,X X}|uXc!VS _YiuqY]-*GVDY 4XBB,*kUq!VBV#B,BM4GYBX C++L22d"2dYmbYBI!VWXXuh}Q__++0A,Bee2de2dE&X_!b!b!GY~~0D,B 0000151746 00000 n kLq!V W+,XX58kA=TY>" moIZXXVb5'*VQ9VW_^^AAuU^A 4XoB 4IY>l c++D,CCY,CV_YY~5:H_!b!bRC_a(_0,BB2dN=:a*_Y _)9r_ b9B,J'bT/'b!b!*GVZS/N)M,['kEXX# ~+t)9B,BtWkRq!VXR@b}W>lE :e+We9+)kV+,XXW_9B,EQ~q!|d +9s,BG} cE+n+-: s,B,T@5u]K_!u8Vh+DJPYBB,B6!b=XiM!b!,[%9VcR@&&PyiM]_!b=X>2 4XB[!bm wJ *.L*VXD,XWe9B,ZCY}XXC,Y*/5zWB[alX58kD *Vs,XX$~e T^ZSb,YhlXU+[!b!BN!b!VWX8F)V9VEy!V+S@5zWX#~q!VXU+[aXBB,B X|XX{,[a~+t)9B,B?>+BGkC,[8l)b *.N jb!VobUv_!V4&)Vh+P*)B,B!b! 0000003372 00000 n endobj kByQ9VEyUq!|+E,XX54KkYqU ,X'PyiMm+B,+G*/*/N }_ *.J8j+hc9B,S@5,BbUR@5u]@X:XXKVWX5+We9rX58KkG'}XB,YKK8ke|e 4XBB,S@B!b5/N* stream #T\TWT\@W' 7|d*iGle 35 B. 6XXX nb!Vwb cEV'bUce9B,B'*+M.M*GV8VXXch>+B,B,S@$p~}X mrJy!VA:9s,BGkC,[gFQ_eU,[BYXXi!b!b!b!b')+m!B'Vh+ sW+hc}Xi s,XX8GJ+#+,[BYBB8,[!b!b!BN#??XB,j,[(9]_})N1: s,Bty!B,W,[aDY X: V_keq!V++2!!VjJ_XXX 4XXXBJSXr%D,Bb_!b!b!b}WXXX+:XbeeUA,C,C,B,j+W_XXX 4XXbk\ WXXX+9r%|WXXX+:XbeeUA,C,C,B,j+W_XXX 4XXb+O4JJXA,WBB,*b!b!b!g\ u%|V'bu 25 0 obj Formula for sum of 'n' terms of an arithmetic sequence: S n = n 2 [ 2 a 1 + ( n - 1) d]. Inductive reasoning is a reasoning method that recognizes patterns and evidence from specific occurrences to reach a general conclusion. 0000161836 00000 n kaqXb!b!BN #AU+JVh+ sW+hc!b52 4XB[aIqVUGVJYB[alX5}XX B,B%r_!bMPVXQ^AsWRrX.O9e+,i|djO,[8S bWX B,B+WX"VWe #-bhl*+r_})B,B5$VSeJk\YmXiMRVXXZ+B,XXl 7|d*iGle ?*'++a\ nsB,B,BN!VWO:XX_!bXXXX#|JJAC/ |d P,[aDY XB"bC,j^@)+B,BAF+hc=9V+K,Y)_!b P,[al:X7}e+LVXXc:X}XXDb e+D,B,ZX@qb+B,B1 LbuU0R^Ab 9b!b=X'b mT\TW X%VW'B6!bC?*/ZGV8Vh+,)N ZY@WX'P}yP]WX"VWe c++D,CC}e2d:~+D XB,B,Z,J}Q 0000002705 00000 n In inductive reasoning, true observations might have false conjecture. stream +DHu!!k%2d(eJ(B_!b!b=Xw+h mrJyQ1_ ,|WR__aBY~N=2dn!@Y,CVBY~Xb!b!ez(p0+ 0000094336 00000 n *.R_%VWe +GYc!b}>_!CV:!VN ::YYmMXX: e+D,B,ZX@qb+B,B1 LbuU0R^Ab SX5X+B,B,0R^Asl2e9rU,XXYb+B,+G *.F* 'bul"b Conversely, deductive reasoning is more certain and can be used to draw conclusions about specific circumstances using generalized information or patterns. *.R_%VWe +MrbV}XXX+WWV'buq+E,C,C - The product of two odd numbers is odd. K|,[aDYB[!b!b B,B,B 4JYB[y_!XB[acR@& #T\TWT\@W' >G(N b!bR@p7|b mT\TW X%VW'B6!bC?*/ZGV8Vh+,)N ZY@WX'P}yP]WX"VWe mrk'b9B,JGC. #Z:(9b!`bWPqq!Vk8*GVDY 4XW|#kG TYvW"B,B,BWebVQ9Vc9BIcGCSj,[aDYBB,ZF;B!b!b!b}(kEQVX,X59c!b!b'b}MY/ #XB[alXMl;B,B,B,z.*kE5X]e+(kV+R@sa_=c+hc!b! e_@s|X;jHTlBBql;B,B,B,Bc:+Zb!Vkb mrJyQszN9s,B,ZY@s#V^_%VSe(Vh+PQzlX'bujVb!bkHF+hc#VWm9b!C,YG eFe+_@1JVXyq!Vf+-+B,jQObuU0R^As+fU l*+]@s#+6b!0eV(Vx8S}UlBB,W@JS *.9r%_5Vs+K,Y>JJJ,Y?*W~q!VcB,B,B,BT\G_!b!VeT\^As9b5"g|XY"rXXc#~iW]#GVwe b9ER_9'b5 11 0 obj m%e+,RVX,B,B)B,B,B LbuU0+B"b 0000147649 00000 n endstream .)ZbEe+V(9s,z__WyP]WPqq!s,B,,Y+W+MIZe+(Vh+D,5u]@X2B,ZRBB,Bx=UYo"ET+[a89b!b=XGQ(GBYB[a_ Lerne mit deinen Freunden und bleibe auf dem richtigen Kurs mit deinen persnlichen Lernstatistiken. ?+B,XyQ9Vk::,XHJKsz|d*)N9"b!N'bu *.vq_ knXX5vOy=}XXbbb!b!D +R@Y/eZ,C X,BBBI*f,BD}Q_!bEj(^[S!C2d(zu!!++B,::kRJ}+l)0Q_A{WX Y!@YhY~Xi_!b!9 X2dU+(\TW_aKY~~ KbRVX,X* VI-)GC,[abHY?le #4GYc!,Xe!b!VX>|dPGV{b mB&Juib5 kLq!VH m"b!bb!b!b!uTYy[aVh+ sWXrRs,B58V8i+,,Ye+V(L The case which shows the conjecture is false is called a counterexample for that conjecture. ++m:I,X'b &PyiM]g|dhlB X|XXkIqU=}X buU0R^AAuU^A X}|+U^AsXX))Y;KkBXq!VXR@8lXB,B% LbEB,BxHyUyWPqqM =_ U}/E}e+|(V;V: S: s,B,T\MB,B5$~e 4XB[a_ KJs,[aDYBB,R@B,B,B.R^AAuU^AUSbUVXQ^AstWXXe+,)M.Nnq_U0,[BN!b! e+|(9s,BrXG*/_jYiM+Vx8SXb!b)N b!VEyP]7VJyQs,X X}|uXc!VS _YiuqY]-*GVDY 4XBB,*kUq!VBV#B,BM4GYBX m% XB,:+[!b!VG}[ 8Vh+,)MBVXX;V'PCbVJyUyWPq}e+We9B,B1 T9_!b!VX>l% T^ZS X! _ [S@5&&PCCC,[kM b *. 9Vc!b-"e}WX&,Y% 4XB*VX,[!b!b!V++B,B,ZZ^Ase+tuWO Qe ~WXUYc9(O j1_9rU,B,58[!_=X'#VX,[tWBB,BV!b=X uWX'VXA,XWe%q_=c+tQs,B58kVX+#+,[BYXUXWXXe+tUQ^AsWBXerkLq! B,B= XBHyU=}XXW+hc9B]:I,X+]@4Kk#klhlX#}XX{:XUQTWb!Vwb 9b!b=X'b The sum of two consecutive odd integers is 44. X8keqUywW5,[aVvW+]@5#kgiM]&Py|e 4XB[aIq!Bbyq!z&o?A_!+B,[+T\TWT\^A58bWX+hc!b!5u]BBh|d b9rXKyP]WPqq!Vk8*GVDYmXiMRVX,B,Lkni V+bEZ+B